A power plant is like a water pump. It generates a pressure (voltage) between its flanges, but water (current) can only flow when the flanges are connected.
If you connect them via a thin pipe (small load / small resistor), water will flow, but the flow is limited by the friction inside the pipe.
With a wider pipe, more water flows.
As long as the pump is strong enough, it will maintain the pressure1). However, if your pipe is too wide, the pressure between the pipes will be smaller than nominal.
Pressure difference times flow is power2), so if there is pressure (voltage) but no flow (current), no power is consumed/produced. Little flow means little power. The pump (plant) only generates the demanded power. A label of 1000W just indicates the maximum output, i.e. the strength. So, nothing is wasted.
(OK, in reality, there are some difficulties. You can not switch on/ off power plants as fast as you need / don't need power, but that's nothing to care about when trying to understand the principles of electricity)
1)This is not true for most pumps in reality, but it describes quite well what happens
2) Here, power = energy / time, i.e. wattage etc.